# Question 2.5.3: An Example of Subsystems: Two Coupled Tanks Consider two bri...

An Example of Subsystems: Two Coupled Tanks

Consider two brine tanks each containing 500 L (liters) of brine connected as shown in Figure 2.5.1. At any time t, the first and the second tank contain x_1(t) and x_2(t) kg of salt, respectively. The brine concentration in each tank is kept uniform by continuous stirring. Brine containing r kg of salt per liter is entering the first tank at a rate of 15 L/min, and fresh water is entering the second tank at a rate of 5 L/min. The incoming brine density r(t) can be changed to regulate the process, so r(t) is an input variable.

Brine is pumped from the first tank to the second one at a rate of 60 L/min and from the second tank to the first one at a rate of 45 L/min. Brine is discharged from the second tank at a rate of 20 L/min.

a. Obtain the differential equations, in terms of x_1 and x_2, that describe the salt content in each tank as a function of time.

b. Obtain the transfer functions X_1(s)/R(s) and X_2(s)/R(s).

c. Suppose that r(t) = 0.2 kg/L. Determine the steady-state values of x_1 and x_2, and estimate how long it will take to reach steady state.

**"Step-by-Step Explanation"**refers to a detailed and sequential breakdown of the solution or reasoning behind the answer. This comprehensive explanation walks through each step of the answer, offering you clarity and understanding.

Our explanations are based on the best information we have, but they may not always be right or fit every situation.

**blue check mark**means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.

Learn more on how we answer questions.